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Foundations of Physics

, Volume 41, Issue 3, pp 516–528 | Cite as

Randomness in Classical Mechanics and Quantum Mechanics

  • Igor V. Volovich
Article

Abstract

The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in the classical mechanics there is fundamental and irreducible randomness. The classical Newtonian trajectory does not have a direct physical meaning since arbitrary real numbers are not observable. There are classical uncertainty relations: Δq>0 and Δp>0, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is always positive (non zero).

A “functional” formulation of classical mechanics was suggested. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton trajectories are computed. An interpretation of quantum mechanics is attempted in which both classical and quantum mechanics contain fundamental randomness. Instead of an ensemble of events one introduces an ensemble of observers.

Keywords

Classical mechanics Quantum mechanics Randomness 

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References

  1. 1.
    Newton, I.: The Mathematical Principles of Natural Philosophy (1729). Newton’s Principles of Natural Philosophy. Dawsons of Pall Mall, London (1968) Google Scholar
  2. 2.
    Boltzmann, L.: Lectures on Gas Theory. University of California Press, Berkeley (1964) Google Scholar
  3. 3.
    Feynman, R.: The Character of Physical Law. Cox and Wyman Ltd., London (1965). A series of lectures recorded by the Ñ at Cornell University USA Google Scholar
  4. 4.
    Landau, L.D., Lifshitz, E.M.: Statistical Physics, Part 1, 3th edn. Pergamon, Elmsford (1980) Google Scholar
  5. 5.
    Ohya, M.: Some aspects of quantum information theory and their applications to irreversible processes. Rep. Math. Phys. 27, 19–47 (1989) MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Prigogine, I.: Les Lois du Chaos. Flammarion, Paris (1994) Google Scholar
  7. 7.
    Accardi, L., Lu, Y.G., Volovich, I.: Quantum Theory and Its Stochastic Limit. Springer, Berlin (2002) MATHGoogle Scholar
  8. 8.
    Kozlov, V.V.: Temperature Equilibrium on Gibbs and Poincaré. Institute for Computer Research, Moscow-Ijevsk (2002). (in Russian) Google Scholar
  9. 9.
    Volovich, I.V.: Number theory as the ultimate physical theory. Preprint No. TH 4781/87, CERN, Geneva (1987). P-Adic Numb. Ultrametric Anal. Appl. 2(1), 77–87 (2010) Google Scholar
  10. 10.
    Volovich, I.V.: p-adic string. Class. Quantum Gravity 4, L83–L87 (1987) CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Manin, Yu.I.: Reflections on arithmetical physics. In: Mathematics as Metaphor: Selected Essays of Yuri I. Manin, pp. 149–155. Am. Math. Soc., Providence (2007) Google Scholar
  12. 12.
    Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: p–Adic Analysis and Mathematical Physics. World Scientific, Singapore (1994) Google Scholar
  13. 13.
    Khrennikov, A.Yu: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic, Dordreht (1997) MATHCrossRefGoogle Scholar
  14. 14.
    Dragovich, B., Khrennikov, A. Yu., Kozyrev, S.V., Volovich, I.V.: On p-adic mathematical physics. P-Adic Numb. Ultrametric Anal. Appl. 1(1), 1–17 (2009) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Volovich, I.V.: Time irreversibility problem and functional formulation of classical mechanics. arXiv:0907.2445
  16. 16.
    Trushechkin, A.S., Volovich, I.V.: Functional classical mechanics and rational numbers. P-Adic Numb. Ultrametric Anal. Appl. 1(4), 361–367 (2009). arXiv:0910.1502 CrossRefMathSciNetGoogle Scholar
  17. 17.
    Volovich, I.V., Trushechkin, A.S.: On quantum compressed states on interval and uncertainty relation for nanoscopic systems. Proc. Steklov Math. Inst. 265, 1–31 (2009) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Vladimirov, V.S.: Equations of Mathematical Physics. Dekker, New York (1971) Google Scholar
  19. 19.
    Pechen, A.N., Volovich, I.V.: Quantum multipole noise and generalized quantum stochastic equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(4), 441–464 (2002) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Khrennikov, A. Yu: Interpretations of Probability. VSP Int. Publ., Utrecht (1999) MATHGoogle Scholar
  21. 21.
    Gozzi, E., Mauro, D.: Quantization as a dimensional reduction phenomenon. AIP Conf. Proc. 844, 158–176 (2006). arXiv:quant-ph/0601209 CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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